 # Are All Sides Equal

The object of this puzzle is to place ten pennies (or other small objects) along the sides of the activity sheet so that each side has exactly the same number. There are several different ways that this can be done. After students have found the solution(s) for ten pennies, they record it. They are then challenged to do the same thing with nine and then eleven pennies. After solving the puzzle, students are asked to study their solutions and then reflect on what they learned in doing the activity.

In the spirit of the Puzzle Corner, students should be allowed to work independently on this puzzle during the week. Then, when enough people have come up with solutions, time should be set aside to let students share their solutions, their methods of arriving or not arriving at a solution, their insights, and any patterns discovered. This sharing and discussion is necessary for Puzzle Corner activities to reach their full potential in the classroom.

I hope that your students will enjoy Are All Sides Equal?

Using the sides of a paper, place ten pennies (or other small objects) so that there are an equal number along each side. Sketch your solution(s) below. Now, do the same with nine pennies, and then with eleven. Sketch your solutions.

Look at the above solutions. What do you notice? What did you learn when doing this activity?

Solution

Click the arrow below to view the solution.

The challenge in Are All Side Equal? was to place nine, ten, and eleven pennies along the edges of a sheet of paper so that there was an equal number along each side. In the solutions below, each circle represents a penny. Two overlapping circles represent two pennies stacked on top of each other, while three overlapping circles represents three stacked pennies. With nine pennies, there are two solutions, one with 3 pennies on a side and one with 4.

There are three solutions with ten pennies. There can be 3, 4, or 5 pennies on each side.

There are three solutions with eleven pennies. There can be 3,4 or 5 pennies on each side.